Definition 1.2 [1]}, [2]} Let \(k\)  be a non-negative integer or infinity. For \(a\in \mathbb {C}\cup \lbrace \infty \rbrace \)  we denote by \(E_{k}(a;f)\)  the set of all \(a\) -points of \(f\) , where an \(a\) -point of multiplicity \(m\)  is counted \(m\)  times if \(m\le k\)  and \(k+1\)  times if \(m>k\) . If \(E_{k}(a;f)=E_{k}(a;g)\) , we say that \(f,g\)  share the value \(a\)  with weight \(k\)  and denote it by \((a,k)\) . The IM and CM sharing corresponds to \((a,0)\)  and \((a,\infty )\)  respectively.

